Mariah Carey -- She Did It Live ... Finally!

Mariah Carey hit the stage at Rockefeller Center for the NBC special -- nearly 24 hours late ... but whatever.

Mariah just performed her classic "All I Want for Christmas is You" for the tree lighting special ... and she did it LIVE ... as promised.

TMZ broke the story ... Mariah pissed off fans and crew members by making them wait in the cold for 3 hours last night when the rest of the special was recorded -- so NBC yanked her from the lineup.

Mimi got a second chance today when the Peacock agreed to tack on her performance ... but ONLY if she did it live. Worth the wait? Watch ...